Amidakuji: count 3-line identity permutations with m,n rungs. a(123456789, 987654321) mod 1234567891. Pure Flow port of the native C helper (modular arithmetic with batch inversion). MOD^2 < 9.2e18 so i64 products are safe.
# Project Euler 837
# Amidakuji: count 3-line identity permutations with m,n rungs.
# a(123456789, 987654321) mod 1234567891.
# Pure Flow port of the native C helper (modular arithmetic with batch
# inversion). MOD^2 < 9.2e18 so i64 products are safe.
import euler.nt { mod_pow }
const MOD: i64 = 1234567891
function mulmod(a0: i64, b0: i64) -> i64 {
let a: i64 = a0 % MOD
let b: i64 = b0 % MOD
return a * b % MOD
}
# Batch-invert consecutive integers start, start+1, ..., start+length-1 mod p.
function invert_consecutive(start: i64, length: i64, out: ptr<i64>) {
if length <= 0 {
return
}
let pref: ptr<i64> = malloc(length * 8)
let mut acc: i64 = 1
for i in 0..length {
acc = acc * ((start + i) % MOD) % MOD
pref[i] = acc
}
let mut inv_acc: i64 = mod_pow(pref[length - 1], MOD - 2, MOD)
let mut i: i64 = length - 1
while i >= 0 {
let prev: i64 = 1
if i != 0 {
prev = pref[i - 1]
}
out[i] = inv_acc * prev % MOD
inv_acc = inv_acc * ((start + i) % MOD) % MOD
i = i - 1
}
free(pref)
}
# Batch-invert a list of values mod p.
function invert_list(vals: ptr<i64>, n: i64, out: ptr<i64>) {
if n == 0 {
return
}
let pref: ptr<i64> = malloc(n * 8)
let mut acc: i64 = 1
for i in 0..n {
acc = acc * vals[i] % MOD
pref[i] = acc
}
let mut inv_acc: i64 = mod_pow(pref[n - 1], MOD - 2, MOD)
let mut i: i64 = n - 1
while i >= 0 {
let prev: i64 = 1
if i != 0 {
prev = pref[i - 1]
}
out[i] = inv_acc * prev % MOD
inv_acc = inv_acc * vals[i] % MOD
i = i - 1
}
free(pref)
}
# C(n,k) mod prime, product formula with batch inversion.
function binom_mod(n: i64, k: i64) -> i64 {
if k < 0 || k > n {
return 0
}
let mut kk: i64 = k
if kk > n - kk {
kk = n - kk
}
if kk == 0 {
return 1
}
let base: i64 = n - kk
let mut res: i64 = 1
let mut start: i64 = 1
let block: i64 = 200000
let invs: ptr<i64> = malloc(block * 8)
while start <= kk {
let mut length: i64 = block
if length > kk - start + 1 {
length = kk - start + 1
}
invert_consecutive(start, length, invs)
let b: i64 = base + start
for i in 0..length {
res = res * ((b + i) % MOD) % MOD
res = res * invs[i] % MOD
}
start = start + length
}
free(invs)
return res
}
function amidakuji_count_mod(m: i64, n: i64) -> i64 {
let L: i64 = m + n
if L % 2 == 1 {
return 0
}
let t: i64 = L / 2
let mut k: i64 = m % 2
let limit: i64 = m
if n < m {
limit = n
}
let mut layout: i64 = 0
if k == 0 {
layout = binom_mod(t, m / 2)
} else {
layout = (t % MOD) * binom_mod(t - 1, (m - 1) / 2) % MOD
}
let inv3: i64 = mod_pow(3, MOD - 2, MOD)
let mut pow2: i64 = 1
if k != 0 {
pow2 = 2
}
let mut sign: i64 = 1
if k != 0 {
sign = MOD - 1
}
let mut total: i64 = 0
let block: i64 = 200000
let nums: ptr<i64> = malloc(block * 8)
let dens: ptr<i64> = malloc(block * 8)
let inv_dens: ptr<i64> = malloc(block * 8)
while k <= limit {
let mut steps: i64 = block
let avail: i64 = ((limit - k) / 2) + 1
if steps > avail {
steps = avail
}
let mut kk: i64 = k
for i in 0..steps {
nums[i] = ((m - kk) % MOD) * ((n - kk) % MOD) % MOD
dens[i] = ((4 * (kk + 1)) % MOD) * ((kk + 2) % MOD) % MOD
kk = kk + 2
}
invert_list(dens, steps, inv_dens)
for i in 0..steps {
let orientations: i64 = ((pow2 + 2 * sign) % MOD) * inv3 % MOD
total = (total + layout * orientations % MOD) % MOD
layout = layout * nums[i] % MOD * inv_dens[i] % MOD
pow2 = pow2 * 4 % MOD
}
k = k + 2 * steps
}
free(nums)
free(dens)
free(inv_dens)
return total
}
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
function malloc(n: i64) -> ptr<void>
}
function main() -> i32 {
printf("%lld\n", amidakuji_count_mod(123456789, 987654321))
return 0
}
Generated C
#include <stdint.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* Flow runtime helpers */
typedef struct flow_temp_node { struct flow_temp_node* next; } flow_temp_node;
static flow_temp_node* flow_temp_head = NULL;
static int flow_temp_atexit_set = 0;
__attribute__((unused)) static void flow_temp_free_all(void) {
while (flow_temp_head) {
flow_temp_node* n = flow_temp_head;
flow_temp_head = n->next;
free(n);
}
}
__attribute__((unused)) static void* flow_temp_alloc(size_t nbytes) {
flow_temp_node* node = (flow_temp_node*)malloc(sizeof(flow_temp_node) + nbytes);
if (!node) return NULL;
node->next = flow_temp_head;
flow_temp_head = node;
if (!flow_temp_atexit_set) {
flow_temp_atexit_set = 1;
atexit(flow_temp_free_all);
}
return (void*)(node + 1);
}
#ifndef FLOW_DIAG
#define FLOW_DIAG(msg) fprintf(stderr, "%s", (msg))
#endif
#ifndef FLOW_LOG
#define FLOW_LOG(fmt, ...) printf(fmt, __VA_ARGS__)
#endif
#ifndef FLOW_LOG_EMPTY
#define FLOW_LOG_EMPTY(fmt) printf(fmt)
#endif
static char* flow_strcat(const char* a, const char* b) {
size_t la = strlen(a ? a : ""), lb = strlen(b ? b : "");
char* r = (char*)flow_temp_alloc(la + lb + 1);
if (!r) return NULL;
if (la) memcpy(r, a, la);
if (lb) memcpy(r + la, b, lb);
r[la + lb] = '\0';
return r;
}
#define __flow_in_arr(arr, val) __extension__ ({ \
int _found = 0; \
size_t _n = sizeof(arr)/sizeof((arr)[0]); \
for (size_t _i = 0; _i < _n; _i++) { \
if ((arr)[_i] == (val)) { _found = 1; break; } \
} _found; })
/* Unified fault handler (MISRA #279) — override with -DFLOW_FAULT_HANDLER=fn */
#ifndef FLOW_FAULT_HANDLER
__attribute__((unused)) static inline void flow_fault_handler(const char* msg) {
fprintf(stderr, "flow: %s\n", msg ? msg : "fault");
abort();
#if defined(__GNUC__) || defined(__clang__)
__builtin_unreachable();
#endif
}
#else
#define flow_fault_handler FLOW_FAULT_HANDLER
#endif
#define flow_div_by_zero_handler() flow_fault_handler("division by zero")
#define flow_shift_ub_handler() flow_fault_handler("invalid shift (amount out of range or left-shift of negative)")
#ifndef FLOW_CHECKED_DIV
#define FLOW_CHECKED_DIV(L, R) (((R) != 0) ? ((L) / (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_MOD
#define FLOW_CHECKED_MOD(L, R) (((R) != 0) ? ((L) % (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHL
#define FLOW_CHECKED_SHL(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull)) && ((L) >= 0)) ? ((L) << (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHR
#define FLOW_CHECKED_SHR(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull))) ? ((L) >> (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#include <math.h>
void* _ui_state = NULL;
static inline float i32_to_f32(int32_t v) { return (float)v; }
/* Host stub for @gpu kernels (device codegen replaces this). */
static inline int32_t gpu_thread_id(void) { return 0; }
int64_t gcd_i64_i64(int64_t a0, int64_t b0);
int64_t lcm_i64_i64(int64_t a, int64_t b);
int64_t isqrt_i64(int64_t n);
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod);
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod);
bool is_prime_i64(int64_t n);
int64_t mulmod_i64_i64(int64_t a0, int64_t b0);
void invert_consecutive_i64_i64_ptr_i64(int64_t start, int64_t length, int64_t* out);
void invert_list_ptr_i64_i64_ptr_i64(int64_t* vals, int64_t n, int64_t* out);
int64_t binom_mod_i64_i64(int64_t n, int64_t k);
int64_t amidakuji_count_mod_i64_i64(int64_t m, int64_t n);
int32_t main(void);
static const int64_t MOD = 1234567891;
int64_t gcd_i64_i64(int64_t a0, int64_t b0) {
int64_t a = a0;
int64_t b = b0;
while (b != 0) {
int64_t t = FLOW_CHECKED_MOD((a), (b));
a = b;
b = t;
}
return a;
}
int64_t lcm_i64_i64(int64_t a, int64_t b) {
if ((a == 0 || b == 0)) {
return 0;
}
return (FLOW_CHECKED_DIV((a), (gcd_i64_i64(a, b))) * b);
}
int64_t isqrt_i64(int64_t n) {
if (n < 2) {
return n;
}
int64_t x = n;
int64_t y = FLOW_CHECKED_DIV(((x + 1)), (2));
while (y < x) {
x = y;
y = FLOW_CHECKED_DIV(((x + FLOW_CHECKED_DIV((n), (x)))), (2));
}
return x;
}
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod) {
int64_t a = FLOW_CHECKED_MOD((a0), (mod));
int64_t b = FLOW_CHECKED_MOD((b0), (mod));
int64_t result = 0;
while (b > 0) {
if (FLOW_CHECKED_MOD((b), (2)) == 1) {
result = FLOW_CHECKED_MOD(((result + a)), (mod));
}
a = FLOW_CHECKED_MOD(((a * 2)), (mod));
b = FLOW_CHECKED_DIV((b), (2));
}
return result;
}
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod) {
if (mod == 1) {
return 0;
}
int64_t result = 1;
int64_t b = FLOW_CHECKED_MOD((base), (mod));
int64_t e = exp;
while (e > 0) {
if (FLOW_CHECKED_MOD((e), (2)) == 1) {
result = mulmod_i64_i64_i64(result, b, mod);
}
b = mulmod_i64_i64_i64(b, b, mod);
e = FLOW_CHECKED_DIV((e), (2));
}
return result;
}
bool is_prime_i64(int64_t n) {
if (n < 2) {
return 0;
}
if (n < 4) {
return 1;
}
if ((FLOW_CHECKED_MOD((n), (2)) == 0 || FLOW_CHECKED_MOD((n), (3)) == 0)) {
return 0;
}
int64_t i = 5;
while ((i * i) <= n) {
if ((FLOW_CHECKED_MOD((n), (i)) == 0 || FLOW_CHECKED_MOD((n), ((i + 2))) == 0)) {
return 0;
}
i = (i + 6);
}
return 1;
}
int64_t mulmod_i64_i64(int64_t a0, int64_t b0) {
int64_t a = FLOW_CHECKED_MOD((a0), (MOD));
int64_t b = FLOW_CHECKED_MOD((b0), (MOD));
return FLOW_CHECKED_MOD(((a * b)), (MOD));
}
void invert_consecutive_i64_i64_ptr_i64(int64_t start, int64_t length, int64_t* out) {
if (length <= 0) {
return;
}
int64_t* pref = (int64_t*)(malloc((length * 8)));
int64_t acc = 1;
int32_t __flow_step_1 = 1;
for (int32_t i = 0; (0 <= length) ? i < length : i > length; i += (0 <= length) ? 1 : -1) {
acc = FLOW_CHECKED_MOD(((acc * FLOW_CHECKED_MOD(((start + i)), (MOD)))), (MOD));
pref[i] = acc;
}
int64_t inv_acc = mod_pow_i64_i64_i64(pref[(length - 1)], (MOD - 2), MOD);
int64_t i = (length - 1);
while (i >= 0) {
int64_t prev = 1;
if (i != 0) {
prev = pref[(i - 1)];
}
out[i] = FLOW_CHECKED_MOD(((inv_acc * prev)), (MOD));
inv_acc = FLOW_CHECKED_MOD(((inv_acc * FLOW_CHECKED_MOD(((start + i)), (MOD)))), (MOD));
i = (i - 1);
}
free(pref);
}
void invert_list_ptr_i64_i64_ptr_i64(int64_t* vals, int64_t n, int64_t* out) {
if (n == 0) {
return;
}
int64_t* pref = (int64_t*)(malloc((n * 8)));
int64_t acc = 1;
int32_t __flow_step_2 = 1;
for (int32_t i = 0; (0 <= n) ? i < n : i > n; i += (0 <= n) ? 1 : -1) {
acc = FLOW_CHECKED_MOD(((acc * vals[i])), (MOD));
pref[i] = acc;
}
int64_t inv_acc = mod_pow_i64_i64_i64(pref[(n - 1)], (MOD - 2), MOD);
int64_t i = (n - 1);
while (i >= 0) {
int64_t prev = 1;
if (i != 0) {
prev = pref[(i - 1)];
}
out[i] = FLOW_CHECKED_MOD(((inv_acc * prev)), (MOD));
inv_acc = FLOW_CHECKED_MOD(((inv_acc * vals[i])), (MOD));
i = (i - 1);
}
free(pref);
}
int64_t binom_mod_i64_i64(int64_t n, int64_t k) {
if ((k < 0 || k > n)) {
return 0;
}
int64_t kk = k;
if (kk > (n - kk)) {
kk = (n - kk);
}
if (kk == 0) {
return 1;
}
int64_t base = (n - kk);
int64_t res = 1;
int64_t start = 1;
int64_t block = 200000;
int64_t* invs = (int64_t*)(malloc((block * 8)));
while (start <= kk) {
int64_t length = block;
if (length > ((kk - start) + 1)) {
length = ((kk - start) + 1);
}
invert_consecutive_i64_i64_ptr_i64(start, length, invs);
int64_t b = (base + start);
int32_t __flow_step_3 = 1;
for (int32_t i = 0; (0 <= length) ? i < length : i > length; i += (0 <= length) ? 1 : -1) {
res = FLOW_CHECKED_MOD(((res * FLOW_CHECKED_MOD(((b + i)), (MOD)))), (MOD));
res = FLOW_CHECKED_MOD(((res * invs[i])), (MOD));
}
start = (start + length);
}
free(invs);
return res;
}
int64_t amidakuji_count_mod_i64_i64(int64_t m, int64_t n) {
int64_t L = (m + n);
if (FLOW_CHECKED_MOD((L), (2)) == 1) {
return 0;
}
int64_t t = FLOW_CHECKED_DIV((L), (2));
int64_t k = FLOW_CHECKED_MOD((m), (2));
int64_t limit = m;
if (n < m) {
limit = n;
}
int64_t layout = 0;
if (k == 0) {
layout = binom_mod_i64_i64(t, FLOW_CHECKED_DIV((m), (2)));
} else {
layout = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((t), (MOD)) * binom_mod_i64_i64((t - 1), FLOW_CHECKED_DIV(((m - 1)), (2))))), (MOD));
}
int64_t inv3 = mod_pow_i64_i64_i64(3, (MOD - 2), MOD);
int64_t pow2 = 1;
if (k != 0) {
pow2 = 2;
}
int64_t sign = 1;
if (k != 0) {
sign = (MOD - 1);
}
int64_t total = 0;
int64_t block = 200000;
int64_t* nums = (int64_t*)(malloc((block * 8)));
int64_t* dens = (int64_t*)(malloc((block * 8)));
int64_t* inv_dens = (int64_t*)(malloc((block * 8)));
while (k <= limit) {
int64_t steps = block;
int64_t avail = (FLOW_CHECKED_DIV(((limit - k)), (2)) + 1);
if (steps > avail) {
steps = avail;
}
int64_t kk = k;
int32_t __flow_step_4 = 1;
for (int32_t i = 0; (0 <= steps) ? i < steps : i > steps; i += (0 <= steps) ? 1 : -1) {
nums[i] = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((m - kk)), (MOD)) * FLOW_CHECKED_MOD(((n - kk)), (MOD)))), (MOD));
dens[i] = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((4 * (kk + 1))), (MOD)) * FLOW_CHECKED_MOD(((kk + 2)), (MOD)))), (MOD));
kk = (kk + 2);
}
invert_list_ptr_i64_i64_ptr_i64(dens, steps, inv_dens);
int32_t __flow_step_5 = 1;
for (int32_t i = 0; (0 <= steps) ? i < steps : i > steps; i += (0 <= steps) ? 1 : -1) {
int64_t orientations = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((pow2 + (2 * sign))), (MOD)) * inv3)), (MOD));
total = FLOW_CHECKED_MOD(((total + FLOW_CHECKED_MOD(((layout * orientations)), (MOD)))), (MOD));
layout = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((layout * nums[i])), (MOD)) * inv_dens[i])), (MOD));
pow2 = FLOW_CHECKED_MOD(((pow2 * 4)), (MOD));
}
k = (k + (2 * steps));
}
free(nums);
free(dens);
free(inv_dens);
return total;
}
int32_t main(void) {
printf("%lld\n", amidakuji_count_mod_i64_i64(123456789, 987654321));
return 0;
}